An introduction to diffractive tomographic microscopyO. Haeberlé1, A. Sentenac2 and H. Giovannini2
1 Laboratoire MIPS/Groupe LabEl, Université de Haute-Alsace, IUT de Mulhouse, 61 rue Albert Camus,68093 Mulhouse cédex - France
2 Institut Fresnel - UMR 6133 CNRS, Bâtiment Fresnel, Campus de St Jérome, 13397 Marseille cé-dex 20 - France
We present the principles of diffractive tomographic microscopy. Contrary to classical transmission mi-croscopy, this technique is based on a coherent, monochromatic, polarized illumination of the specimen,and records the diffracted light in both amplitude and phase using a holographic detection set-up. Com-bined with multiple illuminations of the specimen, a tomography is performed, and a numerical recon-struction of the index of refraction distribution within the specimen is obtained.
The optical microscope has become an invaluable tool for biology thanks to its unique capabilities toimage living specimens in three dimensions, and over long periods (time-lapse microscopy), due to thenon-ionizing nature of light. The fluorescence techniques are particularly appreciated because they allowa specific labelling of cellular structures. However, fluorescent markers may induce unfavourable effectslike photo-toxicity and they do not allow an overall imaging of the sample.As a consequence, among the many techniques, which have been developed, those permitting to observea specimen without the need for specific staining have known a regain of interest in the recent years. Onequotes, for example, the Second-Harmonic Generation microscopy (SHG), the Coherent Anti-StokesRaman Spectroscopy microscopy (CARS) and the conventional transmission microscopy. In transmis-sion microscopy (classical, phase-contrast or Differential Interference Contrast), the image is formed bya complex interaction of the incoherent illuminating light with the specimen. The recorded contrast,while very helpful for morphological studies, does not yield quantitative information on the opto-geometrical characteristics of the sample. In particular, the optical index of refraction distribution withinthe specimen is difficult to reconstruct.
On the contrary, the use of coherent light illumination, combined with an interferometric detection,permits one to record holograms, which encode both the amplitude and phase of the light diffracted bythe specimen. Using an adapted model of diffraction (typically the first order Born approximation), thisso-called holographic microscopy allows for the reconstruction of the specimen index of refraction dis-tribution. When this technique is combined with either a rotation of the specimen or an inclination of theillumination wave, the set of recorded holograms represents a diffractive tomographic acquisition, whichpermits 3-D reconstructions of much better quality.
2. Basics of diffraction and its application to imagingThe aim of this short paragraph is to recall the relation-ship between the opto-geometrical characteris-
tics of an object and its diffracted field. This link is at the core of most imaging systems that use wavesfor probing samples. For simplicity, we will assume the scalar approximation. Accounting for the vecto-rial nature of electromagnetic waves does not change the main lines of the presentation. The interestedreader can complete this rapid introduction by the lecture of any textbook on electromagnetism or wavediffraction, for example [1].
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Fig. 1 A plane wave depicted by its wavevector kinc illu-minates an object characterized by its relative contrast ofpermittivity Δε. The diffracted field is decomposed into asum of plane waves of wavevectors k.
We consider (Figure 1) an object in vacuum defined by its relative permittivity ε(r) and illuminated by amonochromatic incident wave, with wavelength λ = 2πc/ω stemming from a source S(r). Hereafter, theexp(-iωt) dependence is omitted. The total scalar field obeys the Helmholtz equation:
)()()()()( 20
20 rrrrr SEkEkE +εΔ=+Δ , (1)
where k0 is the wavenumber 2π/λ, and the contrast of permittivity Δε = 1-ε. Ιntroducing the Green func-tion of Eq. (1):
G(r) = -exp(ik0r)/4πr , (2)
we obtain the integral equation for the total field,
where Einc is the field generated by S that would exist in absence of the object. The support of the integralin (3) is limited to the geometrical support Ω of the object. When the observation point r is far from theobject, r’2/λ<<r, with r’ in Ω, the diffracted field can be written as:
)(4
)exp()( 0 kr e∂rrikEd −= , rk ˆ0k= , (4)
where
r'r'r'k.r'k dEike )()()exp()( 20 εΔ−= ∫ . (5)
We now suppose that the incident field is a plane wave with wavevector kinc, Einc(r) = Ainc exp(ikinc.r),and that the object is weakly diffracting so that the field inside Ω is close to Einc (Born approximation[2]). In this case, Eq. (5) yields:
)(~),( incinc Ce kkkk −εΔ= , (6)
where C = Ainck02. Equation (6) provides a one to one correspondence between the diffracted far-field
amplitude and the Fourier coefficient of the relative permittivity of the object. It is at the basis of mostfar-field imaging technique such as X-ray diffraction tomography [3], acoustic tomography [4] and, aswill be seen later in section 5, digital holographic microscopy [5]. Note that an expression similar toEq. (6) is obtained in the vectorial case by replacing C by a vector whose direction is given by the pro-jection of the incident field polarization Ainc onto the plane normal to the wavevector k [1].
3. Conventional microscopyIn this section, we first present a brief theoretical description of the functioning of a transmission
microscope based on the paper of Streibl, [6] then we focus on the specific Köhler microscope.We consider the telecentric system described in Fig. 2. The sample is placed before the object focal planeof the objective and the field intensity is recorded on a camera placed at the image focal plane of the tubelens. In this mounting, the object focal plane of the tube lens merges with the image focal plane of theobjective. In most wide-field microscopes, the illuminating source is a thermal lamp, placed at the objectfocal plane of a lens (condenser), which generates a homogeneous illumination that can be seen as thesum of incoherent plane waves impinging on the sample with different directions.
k e(k,kinc) qq
Einc
kinc
Δε
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Fig. 2 Principle of detection of the diffracted wave in a classical transmission microscope
Due to the incoherence properties, the detected intensity on the camera can be considered as the sum ofthe intensities obtained for each incident plane wave. Hence, we will first consider the case where theobject is illuminated by a unique plane wave with wavevector kinc and amplitude Ainc.
The formation of the image in a microscope relies entirely on the analogical Fourier transform that isperformed by the lenses. Indeed, under certain conditions, one can show that the field existing at theimage plane of the lens is proportional to the Fourier transform of the field existing at the object focalplane. Hence, in the set-up presented in Fig. 2, one verifies that the field existing in the image plane ofthe tube lens, namely the CCD plane, will be equal to that existing at the object focal plane of the objec-tive, i. e. very close to the sample. Yet, the field recovery is incomplete due to the loss of informationstemming from propagation and the finite collection cone of the objective. Actually, the imaging systemacts as low-pass filter, symbolized by the pupil placed in the image plane of the objective that cuts all thetransverse Fourier components of the field that are above k0sinθ where θ is the collection angle of theobjective, as shown in Fig. 2, and defines the numerical aperture of the microscope through sinθ =NA(for the sake of simplicity, we consider a microscope in air). More precisely, under the paraxial approxi-mation, the field at the object plane of the objective, EO, can be written as a Fourier integral:
where e(k) is the far-field diffracted amplitude defined in the previous section with k|| the projection of kon the (x,y) plane. In the image plane of the microscope, the field EI can be cast in the form:
where p(u) indicates the filtering function of the imaging set-up. p(u) is a radial function equal to zero ifu > k0NA and one elsewhere. From Eq. (8), it is seen that, by measuring the complex field EI in the im-age plane of a microscope and performing a Fourier transform, one retrieves the far-field amplitude e(k)within the limited solid angle defined by the numerical aperture of the objective. This property will beused in Sections 4 and 5 to build a numerical holographic microscope. In a conventional microscopehowever, one detects the field intensity |EI|2, i. e. the interference between the incident and the diffractedfield1. In this case, the relation-ship between the measured data and the permittivity of the objects ismuch more complex than the one-to-one correspondence pointed out in Eq. (6).
With an incoherent illumination set-up the detected intensity is the sum of |EI|2 for all possible kinc.Long but relatively simple calculations using Eq. (8), Eq. (6) and assuming that the amplitude of thediffracted field is much smaller than that of the incident field, show that the detected intensity is propor-tional to the permittivity contrast Δε of the object, convolved with a point spread function that is differentfor the real and the imaginary part of Δε. This particularity forbids any efficient deconvolution operationfor restoring the image. For absorptive objects and equal illumination and collection solid angle, thepoint spread function P(r||) in the (x,y) plane is given by:
P(r||) = | J1(k0NAr||/k0NA r||) |2 . (9)
1 In dark-field microscopes, the angle of the incident waves is bigger than the collection angle of the objective, i. e. kinc||>k0sinθ. Inthis case, the field that exists at the image plane of the microscope is only the diffracted field.
CCDθ z
r|| k||
incoherentillumination
Tube lens r||objective
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From this formula, one retrieves the classical Abbe criterion that states that two point objects will bedistinguished on the image only if their interdistance is larger than 0.5λ/NA.
The very general analysis provided in this section can be adapted to most microscopes set-up. Amongthose commonly used in biology, one may mention:
- Brightfield microscopy: it is well suited for specimens naturally presenting an important contrastin amplitude. If necessary, dyes may be used to improve the contrast.- Darkfield microscopy: specimens with a low contrast are difficult to image in brigthfield mi-croscopy. Darkfield microscopy is an alternative, in which the specimen is illuminated with lightnormally not collected by the microscope objective. Only light scattered by the specimen is de-tected, and the latter appears brighter than the black background. Rheinberg illumination is a vari-ant, using coloured filters. Oblique illumination is another possible approach.- Polarized microscopy detects variations of optical pathways, related to the thickness and indexof refraction of the specimen. A coloured contrast is produced.- Phase contrast microscopy makes use of the phase difference the diffracted rays experience,when compared to non-diffracted ones. These rays interfere to form an image with higher contrast(a variant is Hoffman illumination).- Differential interference contrast (DIC) microscopy combines feature of polarized microscopyand phase contrast microscopy in order to increase the contrast.
In the following, we stick to brightfield microscopy, and parallel it with diffractive optical tomogra-phy. Figure 3 recalls Fig. 1, but describes in more details the principle of Köhler illumination. The illu-mination source is incoherent, and both spatially and angularly extended. The condenser collects theemitted radiation and concentrates it onto the specimen. The light diffracted by the specimen is collectedby the objective, and refocused so as to form the image onto the detector (here the retina). The benefit ofKöhler illumination is to ensure a uniform illumination of the sample by defocusing as much as possiblethe image of the source. At the same time, the image of the specimen onto the detector is properlyformed. Hence, the illumination source, which is not distinguishable from the light reemitted by thespecimen, forms a uniform background on the detector. It degrades the recorded image essentially byreducing its contrast. Note that, in fluorescence microscopy, the fluorescence wavelength is slightlyshifted with respect to the illumination wavelength (Stokes shift). As a result, a filter permits to collectonly the light emitted by the sample.
Specimen Objective Eye-piece
Source
Object conjugate planes Aperture conjugate planes
Eye-piecediaphragm
Fielddiaphragm
Objective backfocal plane
Condenseraperture
Source
Condenser
Fig. 3 Principle of Köhler illumination. The role of the condenser is to illuminate the specimen underthe largest possible range of angles. The role of the objective is to collect waves, which are diffracted bythe specimen, under the largest possible angles.
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The lateral resolution of a Köhler microscope can be estimated with Eqs. (7,8) by noting that the nu-merical aperture of the condenser NAcond may be different than that of the objective NAobj. One obtains:
Rlat = 0.5λ / (NAcond + NAobj) , (10)
where λ is the average wavelength. This formula points out the role of the condenser in the final resolu-tion. The latter is not determined by the sole numerical aperture of the objective as in fluorescence mi-croscopy.
The limitations of transmission microscopy are mainly the resolution and the difficulty to image low-contrast specimens. The important point is that one directly records an intensity-only image with thelight that has interfered with the specimen. Hence, this contrast is linked to physical quantities (thick-ness, index of refraction, absorption) in a complex manner, which prevents the getting of any quantitativeinformation. Therefore, these techniques are well adapted for dimensional measurements only. Tomo-graphic microscopy, on the contrary, takes advantage of the use of a monochromatic, coherent and po-larized light, to detect both the amplitude and phase of the diffracted wave. The image of the specimen isthen numerically reconstructed in order to simultaneously improve the resolution and record physicalquantities, namely the specimen refractive index distribution.
4. Digital holographic MicroscopyIn this section, we first briefly recall the principles of holography [7,8] and then present its adaptation
to microscopy. For more details about holography, the interested reader may consult Ref. [9].Basically, the principle of holography in the case of a reflective object is the following. A monochro-matic, coherent source produces a light beam, which is split in two beams. One beam is used to illumi-nate the sample, the other being used as a reference beam. The light reflected by the sample is mixedwith the reference beam and sent to a detecting medium (historically, a simple photographic gel), whichrecords the interference pattern between the two beams. The holographic process codes the amplitudeand the phase of the reflected wave into a complex set of interference fringes.
In order to observe the hologram, a decoding procedure has to be applied. In its simplest form, thisdecoding procedure consists in illuminating the obtained hologram with a wave coming from the samesource as the one used to record the hologram. Then, the light diffracted by the hologram yields a threedimensional image of this object. One of the difficulties is that the reconstruction wave must be the sameas the reference wave, in order to avoid deformations in the visible image. Furthermore, a so-called re-versed “twin image” is also obtained in the final reconstruction. If this twin image superimposes with theprimary image, the rendering may be un-interpretable. Hence, to simplify the reading of the hologram,one uses an electronic detector (typically a CCD camera) to record the hologram and to perform numeri-cally the three-dimensional image, by computing the Fresnel diffraction propagation equation.
The adaptation of this principle to microscopy is straightforward. The light diffracted by a micro-scopic specimen, illuminated by a plane wave, is collected by a microscope objective. The role of theobjective is, as in a regular transmission microscope, to collect this diffracted wave within the largestpossible solid angle and to provide a high magnification of the specimen image. This collected wave ismixed with a reference wave and the reference pattern is recorded on a CCD camera. The magnifiedspecimen is then reconstructed numerically from the recorded hologram.
Different configurations based on on-line or on off-line schemes [9-17], working in reflection or intransmission, where the interference pattern is recorded in the Fourier space or in the direct space, havebeen proposed in order to collect useful data on the sample. Another key-point is the measurement of theangular variation of the phase of the field diffracted by the object. Commonly, a phase modulator – typi-cally a piezoelectric retarder or an electro-optic device – is placed along the optical path of either theillumination beam or the reference beam. Varying the retardation, continuously or step-by-step, producesdifferent phase shifted interference patterns that can be recorded on the camera. Data processing tech-niques commonly used in interferometry allow one to determine the phase of the diffracted field from theset of recorded interference patterns.
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Fig. 4 Principle of digital holographic microscopy. The light emitted by the source is split in two parts:an illumination beam and a reference beam. The object O is illuminated either in transmission or in re-flection. A phase modulator can be added either on the path of the illumination beam or on that of the ref-erence beam. The field diffracted by the object O is collected by the microscope objective. In O’ an imageof O is obtained with a strong magnification. The field diffracted in O’ is superimposed coherently to thereference beam. Without Lens L3, the resulting interference pattern is recorded, in far field on the CCDcamera (the camera is placed in the image focal plane of L2 while O’ is in the object focal plane of L2).Adding lens L3 allows one to record an image in the direct space.
Once the complex amplitude of the diffracted field is determined, the opto-geometrical parameters ofthe object are determined numerically by using an appropriated model of diffraction. For example, in thevalidity domain of application of first-order Born approximation, a Fourier transform gives access to thepermittivity distribution of the object, as seen in Eqs (6,8). The aberrations, which may appear with largenumerical aperture microscope objectives, can be compensated numerically [18,19]. The main interest ofdigital holographic microscopy is that it gives, contrary to conventional microscopy, quantitative infor-mation on the object. Moreover a priori information can be included in the reconstruction scheme inorder to reduce the effects of noise and increase the resolution.
5. Diffractive tomographic microscopyIn the previous section, we have seen how digital holographic microscopy yields the three-
dimensional distribution of the sample refractive index from the measurement of its diffracted field (inphase and amplitude) for one illumination. A clear improvement of the resolution can be obtained byusing the concept of synthetic aperture [5, 20-24], namely by using several illuminations, either by ro-tating the sample or changing the incident angle. The main idea is derived from the analysis of Eq. (6)which states that, at infinity (or in the image focal plane of L2 – see Figure 4), the far-field amplitudediffracted in the k direction by an object illuminated under the kinc direction is proportional to the Fouriertransform of the permittivity of the object taken at k-kinc. Hence, by varying the incident and observationangles, one retrieves the Fourier coefficient of the object permittivity within a three-dimensional domainwhose shape and extension completely determines the resolution of the imaging set-up. In a completeconfiguration, where the illumination and observation angles turn all around the object, the accessibleFourier domain is a sphere of radius 2k0 = 4π/λ. In this case, the expected resolution, given by the widthat half-maximum of the inverse Fourier transform of the sphere, is close to 0.35λ in all directions. Yet, inpractice, the obtainable Fourier domain is limited by the numerical aperture of the microscope objectiveto a portion of the Ewald sphere.
We show in Fig. 5 how this Fourier domain can be built for a given experimental configuration. Forthe sake of clarity, Figures 5(a,b,c,d) are 2-D representations in the (kx,kz) plane, (upper row) andFigs. 5(e,f,g,h) are 2-D representations in the (kx,ky) plane (lower row). Note that (kx, ky, kz) are the com-ponents of the wavevector k. In Fig. 5(a) one displays the incident wave, and the circle represents theextremities of the diffracted wavevectors. The set of Fourier components of the diffracted waves that canbe collected by the objective is limited by the numerical aperture (NA) of the microscope objective usedin the detection system, so that only a cap of the Ewald sphere can be recorded.
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Fig. 5 Principle of coherent diffraction tomography: construction of the set of detected waves.
For a given incident wavevector kinc, and an observation direction k, the corresponding spatial fre-quency of the object permittivity is k-kinc (-kinc is indicated by dashed arrow on Figs. 5). Fig. 5(e) depictsthe same process, in the (kx,ky) plane. The set of collected frequencies now describes a dotted disk, whichis centred about the origin.
The idea of synthetic aperture holographic microscopy is to increase the set of recorded Fourier com-ponents, by illuminating the sample with successive waves having different angles of incidence. InFig. 5(b,f) the direction of kinc is changed but the recorded diffracted wavevectors are the same as inFig. 5(a,e). After proper shifting with respect to the incident wavevector extremity, another set of theobject frequencies are detected as shown in Figs. 5(c,g). Figs. 5(d,h) show the different sets of Fouriercomponents that are obtained in the (x,z) and (x,y) planes for normal incidence and for 8 incidences cor-responding to the two maximum polar angles allowed by the numerical aperture of the condenser andfour azimuth angles varying from 0 to 360° every 45° (for the sake of simplicity, we consider that NAcondis equal to NAobj). For example, the upper dashed disk in Fig. 5(h) corresponds to an illumination at 90°azimuth angle. It shows up as the upper dashed arc of circle in Fig. 5(d). When a large number of inci-dences is used, the support of the detected frequencies in the (x,y) plane becomes the disk limited by thebold circle in Fig. 5(h). It is worth noting that the extension of the Fourier domain obtained by varyingthe incidence angle (the bold circle in Fig. 5(h)) is twice that given by a digital holographic microscopewhere sole normal incidence is used (dotted circle in Fig. 5(h)). In the (x,z) plane, the Fourier domainscanned by the diffractive tomographic microscope when a large number of incident angles is used ismore complex, due to the asymmetry of the illumination and detection configuration. It fills the butterflyshape, depicted in Fig. 5(h), which is obtained in a non-trivial way. The right frontier (right dashed arc ofcircle for example) is directly measured in one-shot when using an illumination with the maximum al-lowed polar incidence angle. The upper left short-dash arc of circle however corresponds to the ends ofarc of circles for various incidence angles, and is therefore recorded point by point.
To compare the resolution of digital holography and diffractive tomography more precisely, we depictin Fig. 6 the supports of the detected frequencies that are obtained in a holographic microscope (left) anda diffractive tomographic microscope (right) in the (x,z) plane. From the support extension in Fourierspace, one can estimate the gain in resolution. To be more general, we consider now an immersion me-dium of refractive index n, so that the numerical aperture is now NA = nsinθ.
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Fig. 6 Extension of the detected frequency sets in a holographic microscope (left) and tomographic mi-croscope (right), when NAcond = NAobj.
The lateral and longitudinal dimensions of the frequency support for holographic microscopy are:
€
Δν x,y =2n sinθλ
Δν z =n(1− cosθ)
λ , (11)
respectively, whereas one obtains for tomographic microscopy:
€
Δν x,y =4n sinθλ
Δν z =2n(1− cosθ)
λ , (12)
where θ corresponds to the maximal collection angle of the objective (again for the sake of simplicity,we consider that NAcond is equal to N Aobj). From these frequency supports, one gets the theoretical(Rayleigh) resolution [22] at λ = 633 nm of a holographic microscope with NAcond = 0 and NAobj = 1.4,rxy = 276 nm and rz = 832 nm, to be compared to that of a tomographic microscope with NAcond = 1.4 andNAobj = 1.4, rxy = 138 nm and rz = 416 nm.
In Fig. 7, we plot the Optical Transfer Function (i.e. the Fourier transform of the point spread func-tion) of conventional microscopy, digital holography and synthetic aperture digital holography. It hasbeen seen in section 3 that, under certain assumptions, the image of an incoherent transmission micro-scope is the convolution of the sample permittivity with a point-spread function P given by Eq. (9). Per-forming the inverse Fourier transform of P shows that the microscope acts as a low-pass filter, whosesupport in the Fourier space is the interval [-2NA/λ,+2NA/λ], but whose high frequencies are stronglyattenuated. In holographic microscopy, we have seen that the detection bandwidth is limited to the inter-val [-NA/λ,+NA/λ] in the Fourier space, but these frequencies are detected without attenuation, the trans-fer function being constant over this interval (solid line in Fig. 7). In synthetic aperture holographic mi-croscopy, the detection bandwidth is increased by the angular scanning of the illumination wave to theinterval [-2NA/λ,+2NA/λ]. Thus, one obtains the same detection bandwidth as that of a classical trans-mission microscope, but with a constant transmission (dash line in Fig. 7), thanks to the use of coherentillumination. Consequently, although the images have the same frequency support, synthetic apertureholographic microscopes are expected to present a better resolution than conventional transmission mi-croscopes
νx
OTF
ΝΑ/λ 2ΝΑ/λ
Fig. 7 Comparison of Optical Transfer Functions for incoherent transmission microscopy (dotted line),holographic microscopy (solid line), and coherent diffraction tomography (dashed line). For the sake ofsimplicity, a 1-D representation only is used.
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Note that, from an information processing point of view, the classical transmission microscope withincoherent light is a parallel information processing system, the specimen being simultaneously illumi-nated with all the incidence angles allowed by the condenser, while the diffractive tomographic micro-scope with coherent light is a sequential information processing system. Indeed, it is not possible to si-multaneously illuminate the specimen with coherent light with all the incidences. The incident waveswould form an interference pattern corresponding to the focusing by the condenser, and it would be im-possible to unravel the various Fourier components of the sample from the unique recorded hologram.
Another interesting difference between Köhler illumination in classical transmission microscopy, anddiffractive tomography is the role played by the transmitted specular beam. In classical incoherent mi-croscopy, the non-scattered part of the illumination waves appears as a diffuse luminous background,which renders the observation of the scattered part difficult in the case of weakly diffusing specimens.To improve the observation, one can either use dyes to create a stronger contrast, or use dark-field illu-mination, in which the specimen is illuminated with oblique waves which do not pass through the objec-tive (namely NAcond > NAobj), so that only scattered waves are collected to form a bright image on a darkbackground. In conventional microscopy the non-scattered part of the illumination field is always aninconvenient, which must be dealt with. In diffractive tomography, on the contrary, the presence of thisnon-scattered beam is very helpful since it permits the phase-matching of the holograms, as will be seenin the following.
Fig. 8 presents two images of diatom cells obtained with holographic microscopy and optical diffrac-tion tomography. One clearly observes the better lateral resolution of tomography. The diatom substruc-tures are barely visible in holography, but clearly discriminated in tomography. Furthermore, the longi-tudinal views are radically different. While only characteristic longitudinal diffraction fringes frompunctual objects are visible in holography (there is no discrimination along the z-axis), these fringesdisappear and the horizontal frontiers of the diatom cell are now visible in tomography. This demon-strates the better 3-D imaging capabilities of diffraction tomographic microscopy, compared to classicalholographic microscopy.
x
z
x
z
x
y
x
y
Fig. 8 Diatom cell wall imaged through holographic microscopy (left) and optical diffraction tomography (right)in longitudinal (x-z) and transversal (x-y) views (arrows represent 5µm).
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Hence, it has been shown that the resolution of diffractive tomographic microscopy is, in principle,better than that of a conventional microscope. Yet, this achievement is obtained at a certain expense. Wenow describe the main problems raised by this relatively new technique.
The first issue is to illuminate and observe the sample along various directions. One solution consistsin rotating the sample while keeping the illumination and detection static [25]. This set-up has the ad-vantage that the optical interferometer has no moving parts, which is favourable in terms of vibrations oralignments. However, keeping a precise rotation at the microscopic scale, compatible with interferomet-ric measurements can be very difficult. Furthermore, this set-up necessitates placing the observed samplein a rotating microcapillary. This procedure has proven to work well for individual cells or pollen grains,for example, but is not favoured by biologists, who often prefer to handle biological samples between aglass slide and a coverglass. Furthermore, using a rotating capillary oblige to use a longer working dis-tance objective with a lower numerical aperture, at the detriment of the resolution. It is therefore prefer-able to keep the specimen static. For technical reasons, it is then easier to rotate the illumination than torotate the detection, which would need to rotate both the reference beam and the collected diffractedbeam, which have to interfere [22].
The second issue is the retrieval, for numerous successive illuminations, of the amplitude and phase ofthe diffracted field from an interference pattern. We assume that the optical path difference between theillumination and the reference beams is constant during the acquisition of each hologram correspondingto one incidence. However, between two illuminations, the phase between the illumination and the refer-ence beams may change due to thermal/mechanical drifts or simply because the optical paths of the illu-mination beam are different for the various incidence angles. This variation leads to a phase shift be-tween the holograms. For retrieving the complex Fourier components of the sample from the synthetichologram, one has to compensate for these phase shifts by phase-matching the holograms. This can bedone by considering the shared spatial frequencies detected at two different incidence angles. Commonspatial frequencies correspond to the condition : k-kinc=k’-k’inc, where kinc and k’inc are the two incidentwavevectors, k and k’ are two diffracted wavevector. Within the validity of Born approximation, Eq. (6),the phases of the diffracted amplitude e(k,kinc) and e(k’,k’inc) are to be equal. If this is not the case, aphase matching of the two holograms is performed by adding a constant phase to one of the holograms.This procedure, adopted for all the recorded holograms, compensates for the random phase drifts. Othertechniques based on the measurement of the phase between the reference beam and the non-scatteredlight (i.e. the specular transmitted beam), have also been proposed [22]. A frequency analysis and a nu-merical post-processing of the data [26] can also be used.
Note that the detection of the diffracted field can be performed in different ways, either in the Fourierplane [22,24], (in the image focal plane of L2 in Fig. 4) or in the image plane [23] (by adding the lens L3in Fig. 4). The main drawback of the measurements in the Fourier space comes from the fact that thespecular beam (the non-scattered light) is focused onto a small surface of the camera and may saturatethe detector. Attenuating too strongly the incident beam is not possible because it would lead to a lowsignal-to-noise ratio in the regions of low intensity on the camera. For solving this problem, one can usea variable density on the illumination beam. For each incidence angle, two holograms can be recordedand processed: one at low incident intensity, which serves to measure the incident beam, together withthe low frequency part of the spectrum, another at strong intensity, which serves to measure the highfrequency components [22]. Data fusion techniques then permit to reconstruct the spectrum withoutsaturation near the frequency origin. Measurements in the image plane avoid this drawback. In this case,for each incidence, a Fourier transform is performed to retrieve the hologram in the Fourier space as inthe previous case. The calculated holograms are then processed in order to obtain the synthetic hologram.
Another important point of this technique is the calibration of the diffracted intensity. The latter isnecessary if one wants to obtain quantitative information on the sample [26]. For this purpose, the fielddiffracted by a sample with known opto-geometrical parameters has to be measured. A comparison withthe results predicted by theory allows one to calibrate the system by normalizing the value of the dif-fracted intensities.
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6. Further developmentsFrom section 2, we have seen that, under the Born approximation, the permittivity distribution of the
object can be recovered, from the measured complex amplitude of the diffracted field, thanks to a Fouriertransform. Depending on the opto-geometrical parameters of the object, other models of diffraction canbe used. For example, the profile of a reflecting surface can be obtained by using Fraunhofer approxima-tion. In this case, the complex amplitude of the diffracted field is given by,
IIIIIIIIII krkkrkk diRe incinc ]).-(exp[)(),( ∫∫= , (13)
where R(r||)=R0exp[2k0h(r||)] is the reflectivity of the object with R0 being the reflection coefficient of thesurface and h(r||) the local height of the object. Hence, calculating the Fourier transform of the diffractedfield gives access to h(r||).
Approximate models of diffraction allow one to retrieve, with simple inversion schemes, the opto-geometrical parameters of the object from its diffracted far-field. Most of them are based on the singlescattering approximation, which is valid essentially for weakly scattering samples. Yet, in certain casesthey do not describe properly the diffraction phenomenon. Typically, when the objects are greater thanthe illuminating wavelength and for strong permittivity contrasts, one needs to account for multiplescattering. In this case, iterative reconstruction algorithms based on a rigorous theory of diffraction,which basically, requires to solve Eq. (3), have to be used [27-29]. These algorithms are widely used forimagery applications in the radiofrequency domain where single scattering assumption does not oftenhold. Note that, when multiple scattering occurs, the diffracted far-field may gives access to informationcorresponding to spatial frequencies of the field that do not propagate (namely the evanescent compo-nents). Indeed, multiple scattering produces a coupling between the spatial frequencies and e(k,kinc) is nomore a function of k-kinc. For this reason, a resolution better than the Rayleigh-Abbe limit can be ob-tained [27-29].
Another way to improve the resolution is to increase the value of the spatial frequency of the incidentfield in order to extend the available Fourier support of the sample. This can be done, for example, byimmerging the sample and the objective into a medium of high refractive index, or by illuminating thesample in total internal reflection, with Total Internal Reflection Tomography, (TIRT) [27-29]. Thisconfiguration permits theoretically an improvement of the resolution by a factor of n, where n is therefractive index of the prism used to illuminate the sample. However, since the evanescent field thatilluminates the object decreases exponentially as one moves away from the substrate, its application isrestricted to surface imaging. Recently [29], it has been proposed to deposit the object onto a sub-lambdagrating in order to increase the spatial frequency of the incident field beyond the value given by theavailable indices of refraction. Indeed, the field above the grating presents arbitrary large spatial frequen-cies. It has been shown theoretically that the resolution of a grating-assisted microscope can be muchbetter than that of TIRT and well beyond the Abbe limit.
Another application concerns the statistical characteristics of random objects such as rough surfaces orheterogeneous films. In these cases, the measurement of the phase of the diffracted field gives access toinformation that cannot be obtained from intensity measurements. In the case of rough surfaces, for ex-ample, it yields the height distribution probability density, which is a more precise characterization of thesurface than the power spectral density function classically obtained with intensity measurements.
7. ConclusionTomographic microscopy using coherent illumination is a promising technique, because it is expected
to have a much better resolution than conventional microscopes with equal numerical aperture. Moreoverit gives access to the distribution of the index of refraction within a specimen, a quantity, which is not orvery hardly accessible to present microscopes. Yet, using coherent illumination presents some chal-lenges. In particular, one must use a sequential acquisition of the data, and a post-acquisition numericalreconstruction of the image, which limits the speed of acquisition. Moreover, an absolute calibration of
Modern Research and Educational Topics in Microscopy
the system is necessary to obtain quantitative information on the index of refraction and still remains tobe done.
In this review, we have considered only transmission set-ups, but this technique can also be adapted toreflection microscopy. Detecting both the transmitted and the reflected field would pave the way to ahigh-resolution isotropic 3-D imaging tool for non-labelled transparent specimens. The progress in thisdomain are continuous, thanks to several research groups, and one can expect to see, in the future, dif-fractive tomographic microscopy (and its simplified version: holographic microscopy) become a routinetool for biology, in complement to other, more-established techniques.
Acknowledgements The authors gratefully acknowledge Patrick Chaumet and Kamal Belkebir for fruitful discus-sions, and Matthieu Debailleuil, Vincent Georges, Vincent Lauer and Bertrand Simon for their enlightening contri-butions and for providing the diatome images to illustrate this article. They also thank Vincent Lauer for interestingdiscussions about the tomographic technique he developed.
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